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Review of ANOVA & Inferences About The Pearson Correlation Coefficient. Heibatollah Baghi, and Mastee Badii. Review of ANOVA (1). Review of ANOVA (2). Review of ANOVA (3). S.V. SS DF MS F c F α -------------- ------ ------ ------ ----- ----- Systematic Effect 70 2 35 9.13 3.88

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## Review of ANOVA & Inferences About The Pearson Correlation Coefficient

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**Review of ANOVA & Inferences About The Pearson Correlation**Coefficient Heibatollah Baghi, and Mastee Badii**Review of ANOVA (3)**S.V. SS DF MS Fc Fα -------------- ------ ------ ------ ----- ----- Systematic Effect 70 2 35 9.13 3.88 Random Effect 46 12 3.83 ------- ---------- ----- ----- ------- Total 116 14**Practical Significance or Effect Size in ANOVA**• Statistical significance does not provide information about the effect size in ANOVA. • The index of effect size is η2 (eta-squared) • η2 = SSB / SST or η2 = 70/116 = .60 • 60 % of the variability in stress scores is explained by different treatments.**Practical Significance or Effect Size in ANOVA, Continued**Source SS DF MS Fc Fαη2 --------- ------ ------ ----- ------- ----- ---- Between 70 2 35.0 9.13 3.88 .60 Within 46 12 3.83 ------- ------ ---- ----- ----- ------- Total 116 14**Sample Size in ANOVA**• To estimate the minimum sample size needed in ANOVA, you need to do the power analysis. • Given the: α = .05, effect size = .10, and a power ( 1- beta) of .80, 30 subjects per group would be needed. (Refer to Table 7-7, page 178).**Inferences About The Pearson Correlation Coefficient**Refer to Session 5GPA and SAT Example**Population of visual acuity and neck size “scores”**ρ=0 Sample 1 Sample 2 Sample 3 Etc r = -0.8 r = +.15 r = +.02 Relative Frequency 0 µr r: The development of a sampling distribution of sample v:**Steps in Test of Hypothesis**• Determine the appropriate test • Establish the level of significance:α • Determine whether to use a one tail or two tail test • Calculate the test statistic • Determine the degree of freedom • Compare computed test statistic against a tabled/critical value Same as Before**1. Determine the Appropriate Test**• Check assumptions: • Both independent and dependent variable (X,Y) are measured on an interval or ratio level. • Pearson’s r is suitable for detecting linear relationships between two variables and not appropriate as an index of curvilinear relationships. • The variables are bivariate normal (scores for variable X are normally distributed for each value of variable Y, and vice versa) • Scores must be homoscedastic (for each value of X, the variability of the Y scores must be about the same) • Pearson’s r is robust with respect to the last two specially when sample size is large**2. Establish Level of Significance**• α is a predetermined value • The convention • α = .05 • α = .01 • α = .001**3. Determine Whether to Use a One or Two Tailed Test**• H0 : ρXY = 0 • Ha : ρXY ≠ 0 • Ha : ρXY > or < 0 Two Tailed Test if no direction is specified One Tailed Test if direction is specified**5. Determine Degrees of Freedom**For Pearson’s r df = N – 2**6. Compare the Computed Test Statistic Against a Tabled**Value • α = .05 • Identify the Region (s) of Rejection. • Look up tα corresponding to degrees of freedom**Example of Correlations Between SAT and GPA scores**• Formulate the Statistical Hypotheses. • Ho : ρXY = 0 Ha : ρXY ≠ 0 • α = 0.05 • Collect a sample of data, n = 12**Check Significance**• Identify the Region (s) of Rejection. • tα = 2.228 • Make Statistical Decision and Form Conclusion. • tc < tα Fail to reject Ho • p-value = 0.095 > α = 0.05 Fail to reject Ho • Or use Table B-6: rc = 0.50 < rα =.576 Fail to reject Ho**Practical Significance in Pearson r**• Judge the practical significance or the magnitude of r within the context of what you would expect to find, based on reason and prior studies. • The magnitude of r is expressed in terms of r2 or the coefficient of determination. • In our example, r2 is .50 2 = .25 (The proportion of variance that is shared by the two variables).**Sample Size in Pearson r**• To estimate the minimum sample size needed in r, you need to do the power analysis. For example, Given the: α = .05, effect size (population r orρ) = 0.20, and a power of .80, 197 subjects would be needed. (Refer to Table 9-1). Note: [ρ= .10 (small), ρ=.30 (medium), ρ =.50 (large)]**Magnitude of Correlations**• ρ = .10 (small) • ρ = .30 (medium) • ρ = .50 (large)**Factors Influencing the Pearson r**• Linearity. To the extent that a bivariate distribution departs from normality, correlation will be lower. • Outliers. Discrepant data points affect the magnitude of the correlation. • Restriction of Range. Restricted variation in either Y or X will result in a lower correlation. • Unreliable Measures will results in a lower correlation.**Take Home Lesson**How to calculate correlation and test if it is different from a constant

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